Representations of the Symmetric Group

Abstract

In this chapter we construct all the irreducible representations of the symmetric group. We know that the number of such representations is equal to the number of conjugacy classes (Proposition 1.10.1), which in the case of S _n is the number of partitions of n. It may not be obvious how to associate an irreducible with each partition λ = (λ₁, λ₂,...., λ_l), but it is easy to find a corresponding subgroup S _λ that is an isomorphic copy of S _λl x S_λ2 x · · · x S _λl, inside S _n . We can now produce the right number of representations by inducing the trivial representation on each Sλ up to S _n .